As concerns about PG&E Corp.’s financial health spiral, California Gov. Gavin Newsom (D) signaled that he’ll deal with the troubles plaguing the state’s largest utility as one of his first tasks in office.

In his only public comments on the company since his inauguration, Newsom said at a press conference that he’s working on PG&E matters “in real time” and is talking with state energy regulators, members of the previous administration, and incoming staff “to address the solvency” of the...